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A000354
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Expansion of e^{-x}/(1-2*x).
(Formerly M3957 N1631)
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13
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1, 1, 5, 29, 233, 2329, 27949, 391285, 6260561, 112690097, 2253801941, 49583642701, 1190007424825, 30940193045449, 866325405272573, 25989762158177189, 831672389061670049, 28276861228096781665, 1017967004211484139941
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) is the permanent of the n X n matrix with 1's on the diagonal and 2's elsewhere. - Yuval Dekel, Nov 01 2003. Compare A157142.
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 13 2009: (Start)
Starting with offset 1 = Lim_{k->inf.} M^k, where M = a tridiagonal matrix
with (1,0,0,0,...) in the main diagonal, (1,3,5,7,...) in the subdiagonal
and (2,4,6,8,...) in the subsubdiagonal. (End)
a(n) is also the number of (n-1)-dimensional facet derangements for the n-dimensional hypercube. [From Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009]
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REFERENCES
| Gary Gordon and Elizabeth McMahon, arXiv:0906.4253 : Moving faces to other places: Facet derangements.
G. Gordon and E. McMahon, Moving faces to other places: facet derangements, Amer. Math. Monthly, 117 (2010), 865-88.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
E. Lucas, Th\'{e}orie des Nombres. Gauthier-Villars, Paris, 1891, Vol. 1, p. 223.
Plouffe Simon, Exact formulas for integer sequences
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FORMULA
| Inverse binomial transform of double factorials A000165 - Paul Barry (pbarry(AT)wit.ie), May 26 2003
a(n)=sum{k=0..n, (-1)^(n+k)C(n, k)k!2^k } - Paul Barry (pbarry(AT)wit.ie), May 26 2003
a(n)= Sum(k=0..n, A008290(n, k)*2^(n-k)) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 13 2003
a(n) = 2n*a(n-1)+(-1)^n, n>0, a(0)=1. - Paul Barry (pbarry(AT)wit.ie), Aug 26 2004
a(n) = (2n-1)a(n-1) + (2n-2)a(n-2) [From Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009]
a(n) = round(2^n*n!/exp(1/2)) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Jul 06 2010]
Contribution from Groux Roland Jan 17 201: (Start)
a(n)=(1/(2*sqrt(exp(1))))*int(exp(-x/2)*x^n,x=-1..infinity)
sum(1/(k!*2^(k+1)*(n+k+1),k=0..infinity)=(-1)^n*[a(n)*sqrt(exp(1))-2^n*n!] (End)
a(n)={2^n*n!/exp(1/2)}, x >= 0 and {x} is the nearest integer function. Simon Plouffe, March 1993.
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MAPLE
| BB := (x, k)->k!*sum(sum(x^j/((k-j)!^2*j!), j=1..k), m=1..k): R := (x, n, k)->BB(x, k)^n: f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k): > seq(abs(f(0, n, 2)/2!^n), n=0..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 26 2007
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MATHEMATICA
| Table[ Gamma[ n, -1/2 ]*2^(n-1)/Exp[ 1/2 ], {n, 1, 24} ]; FunctionExpand[ % ]
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CROSSREFS
| Cf. A061714.
Cf. A008290.
Sequence in context: A057623 A087662 A113012 * A103815 A134752 A144015
Adjacent sequences: A000351 A000352 A000353 * A000355 A000356 A000357
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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